(1) (via Peter Winkler) Here's a (seemingly) new hat puzzle for you: 100 prisoners will be fitted with red or blue hats according to fair coinflips. Then the lights are turned on; each prisoner sees every other hat color and must write down a guess for his own. If a majority (at least 51) get their own hat color right, all the prisoners will be freed. As usual the prisoners can't communicate once the hats are placed, but have time to strategize beforehand. Can they achieve a 90% probability of being freed? How about 95%? (2) (Johan Wästlund) I'll just throw in here a question of (what turns out to be) similar flavor: Suppose you play a type of one-person tic-tac-toe where in every "move" you select a set of unoccupied squares, and then flip a coin to decide whether you get all those squares, or your "opponent" gets them. What winning probability can you achieve? -- Thane Plambeck tplambeck@gmail.com http://counterwave.com/